Solved Examples on 2^k Factorial Experiment lecture - 28

 Examples on 2^k Factorial Experiment 

lecture - 28 

Example: A 4 X 4 Latin square was laid out to investigate the effect of 4 treatment combinations (nitrogen at levels n0 and n1 and phosphates at levels P0 and P1) on the yield of potatoes. The field plans together with plot yield are as follows:

Perform complete analysis of the data and give your interpretation of the results.

Solution: We setup our hypotheses as:

H0 : τ1 = τ2 = τ3 = τ4 Vs. H1 : τ1 ≠ τ2 ≠ τ3 ≠ τ4 α

H01: The effect of nitrogen on yield is insignificant 

                                    Vs. 

H11: The effect of nitrogen on yield is significant

H02: The effect of phosphorus on yield is insignificant 

                                 Vs. 

H12: The effect of phosphorus on yield is significant

H03: The effect of nitrogen & phosphorus combination on yield is insignificant.

                                Vs.

H13: The effect of nitrogen & phosphorus combination on yield is significant.

For treatment:

Partition of treatment sum of squares by using Yates Algorithm

Treatment Combination	Yield	Column – I	Column - II	Effects
(1)	50	132	318
n	82	186	74	342.25
P	72	32	54	182.25
np	114	42	10	6.25

ANOVA Table:

Remarks: H0 is rejected and the effect of 4 treatment combination on potatoes yield is significant.

H01 and H02 is also significant and the individual effects of nitrogen and phosphorus are significant. 

Example: 

An experiment is performed in order to investigate the effect of concentration of a reactant and the presence of the catalyst of the reaction time of a chemical process. Let the two levels of the reactant concentration be 15 % & 25 %. The catalyst has two levels high level denoting the presence of the catalyst and low level denoting the absence with three replicates. The data of the experiment are shown in the table below: 

Treatment Combination	Replicate
	I	II	III
A low, B low	28	25	27
A high, B low	36	32	32
A low, B high	18	19	23
A high, B high	31	30	29

Test the effects of reactant concentration and catalyst and their interaction.

Solution: 

i. We setup our hypotheses as:

H0 : τ1 =τ2 = τ3 = τ4 Vs. H1 : τ1 ≠τ2 ≠ τ3 ≠ τ4 

H01 : Aj = 0 Vs. H11 : Aj ≠ 0

H02 : Bk = 0 Vs. H12 : Bk ≠ 0

H03 : (AB)jk = 0 Vs. H13 : (AB)jk ≠ 0

ii. The significance level; α = 0.05

iii. The test statistic:

vi.  Critical Regions:

Reject H0, When F .= > F0.05(3, 8) = 4. 07

Reject H01, H02, H03, when F .= > F0.05(3, 8) = 5. 12

v. Computation:

Treatment Combination	Total	Column I	Column II	Effects
(1)	80	180	330
a	100	150	50	208.33
b	60	20	-          30	75
ab	90	30	10	8.33

ANOVA Table:

vi. Remarks

       The F3 calculated value falls in the acceptance area . It is concluded that the interaction effect is insignificant.

Example: 

i.               The following output was obtained from a computer program that performed a two-factor ANOVA on a factorial experiment.

SV	d.f	S. S	M . S	F
A	1	0.322	?	?
B	?	80.554	40.2771	4.59
AB	?	?	?	?
Error	12	105.327	8.7773
Total	17	231.551

a.    Fill in the blanks in the ANOVA table

b. How many levels were used for factor B?

c.  How many replicates of the experiment were performed?

d.  What conclusions would you draw about this experiment?

Solution:

Solution:

a.   MS for factor A = 0.322 / 1 = 0.322

d.f for factor B = 1

d.f for factor AB = 1

SSAB = SSTotal  -  (SSE + SSA+ SSB)

SSAB = 231.551 – (105.327 + 0.322 + 80.554)

SSAB = 45.348

MSAB = SSAB / df

MSAB = 45.348 / 1

MSAB = 45.348

SV	d.f	S. S	M . S	F
A	1	0.322	0.322	0.0366
B	1	80.554	40.2771	4.59
AB	1	45.348	45.348	5.166
Error	12	105.327	8.7773
Total	17	231.551

b. The experiment is 2^2 factorial experiment, so two levels are used for factor B.

c. d.f for Treatment = 3

    d.f for Error = 12

    d.f for Total = 17

    d.f for Replicate or Row = 17 - 12 - 3

    d.f for Replicate or Row = 2

so, the number of Replicates = 3

d. 

SV	d.f	S. S	M . S	F	F tab
A	1	0.322	0.322	0.0366	3.18
B	1	80.554	40.2771	4.59	3.18
AB	1	45.348	45.348	5.166	3.18
Error	12	105.327	8.7773
Total	17	231.551

The interaction effect of AB and effect of factor B are significant.

Example: The ANOVA table of RCBD Factorial experiment is given below:

SV	df	SS	MS	F	P value
A	1	36.00	?	?	?
B	?	?	0.75	?	?
AB	1	12.00	?	?	?
Error	6	7.50	?
Total	11	56.25

Answer the following questions:

(a).  The form of the above factorial experiment…...

(b). The statistical model for the factorial experiment…...

(c). Find the missing values and complete the ANOVA table.

(d). Determine P values.

(e). interpret the results.

(f). How many blocks were used

Answers:

(a). 2^2 factorial experiment

(b). Statistical model:

Yijk = μ + Aj + Bk + (AB)jk + eijk

(c). Complete ANOVA

(d) using the table

SV	df	SS	MS	F	P value
A	1	36.00	36.00	28.8	0.010
B	1	0.75	0.75	0.6	0.100
AB	1	12.00	12.00	9.6	0.010
Error	6	7.50	1.25
Total	11	56.25

Example:

A full-factorial design is being used to help with development of a prospective 3-D printing process for a ceramic scaffold material, in an attempt to improve the compressive strength. Three factors, each with two levels, have been tested, producing the results shown in the tables below:

Factor	Levels
	0	1
A: Powder size (nm)	50	100
B: Layer thickness (mm)	0.3	0.5
C: Scan speed (mm/min)	100	300

 Write down appropriate statistical model of the above experiment. Using ANOVA, determine the effect of powder size, Layer thickness, Scan speed, and the interaction between them.

Solution:The experiment consists 3 factors (i.e. A, B, C) and each factor have two levels, the following statistical model is used to represent the mentioned experiment.

Yijkl = μ + Aj + Bk + Cl + (AB)jk + (AC)jl + (BC)kl + (ABC) jkl + eijkl

	Treatment Combination of 3^2 Factorial Experiment
	a0b0c0	a1b0c0	a0b1c0	a1b1c0	a0b0c1	a1b0c1	a0b1c1	a1b1c1
	(1)	a	b	ab	c	ac	bc	abc
	50	100	50	100	50	100	50	100
	0.3	0.3	0.5	0.5	0.3	0.3	0.5	0.5
	100	100	100	100	300	300	300	300
TC	150.3	200.3	150.5	200.5	350.3	400.3	350.5	400.5

Now using Yates Method

 

TC	Total	Column I	Column II	Column III	Effects
(1)	150.3	350.6	701.6	2203.2
a	200.3	351.0	1501.6	200	5000
b	150.5	750.6	100	0.8	0.08
ab	200.5	751.0	100	0	0
c	350.3	50	0.4	800	8000
ac	400.3	50	0.4	0	0
bc	350.5	50	0	0	0
abc	400.5	50	0	0	0

The interaction effects are insignificant and factor A, C are significant.

• Read More:  2^k Practice Questions